GEOMETRY COORDINATE GEOMETRY Proofs Name Period 1
Coordinate Geometry Proofs Slope: We use slope to show parallel lines and perpendicular lines. Parallel Lines have the same slope Perpendicular Lines have slopes that are negative reciprocals of each other. Midpoint: We use midpoint to show that lines bisect each other. Lines With the same midpoint bisect each other Midpoint Formula: mid x1 x y y, 2 2 2 1 2 Distance: We use distance to show line segments are equal. You can use the Pythagorean Theorem or the formula: 2
Proving a triangle is a right triangle Method: Show two sides of the triangle are perpendicular by demonstrating their slopes are opposite reciprocals. Example 1: Using Geometer s Sketchpad Show: ABC is an isosceles right triangle. Question: Formula: Work: Statement: 3
Practice 1: The vertices of triangle JEN are J(2,10), E(6,4), and N(12,8). Use coordinate geometry to prove that Jen is an isosceles right triangle. Practice 2: Prove that the polygon with coordinates A(1, 1), B(4, 5), and C(4, 1) is a right triangle. 4
Practice 3: Using Geometer s Sketchpad Show that the triangles with the following vertices are isosceles. SUMMARY 5
Homework 1. Prove the polygon with the given coordinates is a right triangle. 2. Show that the triangles with the following vertices are isosceles. 6
3. 7
Proving a Quadrilateral is a Parallelogram Method 1: Show that the diagonals bisect each other by showing the midpoints of the diagonals are the same Method 2: Show both pairs of opposite sides are parallel by showing they have equal slopes. Method 3: Show both pairs of opposite sides are equal by using distance. Method 4: Show one pair of sides is both parallel and equal. Example Model Problem Prove that the quadrilateral with the coordinates P(0, 2), Q(4, 8), R(7, 6) and S(3, 0) is a parallelogram. Question: Formula: Work: Statement: 8
Practice 1. Prove that the quadrilateral with the coordinates L(-2,3), M(4,3), N(2,-2) and O(-4,-2) is a parallelogram. 2. Prove that the quadrilateral with the coordinates P(1,1), Q(2,4), R(5,6) and S(4,3) is a parallelogram. 9
3. Prove that the quadrilateral with the coordinates R(3,2), S(6,2), T(0,-2) and U(-3,-2) is a parallelogram. Summary 10
Homework 11
Proving a Quadrilateral is a Rectangle Prove that it is a parallelogram first, then: Method: Show that the diagonals are congruent. Example Model Problem 1. Prove a quadrilateral with vertices G(0,5), H(6,9), I(8,6) and J(2,2) is a rectangle. 12
Practice 1. The vertices of quadrilateral COAT are C(0,0), O(5,0), A(5,2) and T(0,2). Prove that COAT is a rectangle. 13
Practice 2 Summary 14
Homework 15
Proving a Quadrilateral is a Rhombus Method: Prove that all four sides are equal. Example Model Problem Prove that a quadrilateral with the vertices A(-2,3), B(2,6), C(7,6) and D(3,3) is a rhombus. Statement: 16
Practice 1. Prove that the quadrilateral with the vertices D(2,1), A(6,-2), V(10,1) and E(6,4) is a rhombus. 2. 17
Summary Homework 1. 18
2. Prove that quadrilateral TIME with the vertices T(1,1), I(5,3), M(7,7), and E(3,5) is a rhombus. 19
Proving that a Quadrilateral is a Square There are many ways to do this. Prove that the quadrilateral is a rectangle and a rhombus. Example Model Problem 1. Prove that the quadrilateral with vertices A(0,0), B(4,3), C(7,-1) and D(3,-4) is a square. Statement: 20
Practice 1. Prove that the quadrilateral with vertices A(2,2), B(5,-2), C(9,1) and D(6,5) is a square. 21
2. Summary 22
Homework 23
Proving a Quadrilateral is a Trapezoid Show one pair of sides are parallel (same slope) and one pair of sides are not parallel (different slopes). Example Model Problem Prove that DEFG a trapezoid with coordinates D(-4,0), E(0,1), F(4,-1) and G(-4,-3). 24
Proving a Quadrilateral is an Isosceles Trapezoid Prove that it is a trapezoid first, then: Method 1: Prove the diagonals are congruent using distance. Method 2: Prove that the pair of non parallel sides are equal. Example Model Problem 1. Prove that quadrilateral BCDE with the vertices B(0,4), C(3,1), D(3, -5), and E(0,-8) is an isosceles trapezoid. Question: Formula: Work: Statement: 25
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SUMMARY 27
Homework Quadrilateral KATE has vertices K(1,5), A(4,7), T(7,3), and E(1, 1). Prove that KATE is a trapezoid. 28
Practice with Coordinate Proofs 1. The vertices of ABC are A(3,-3), B(5,3) and C(1,1). Prove by coordinate geometry that ABC is an isosceles right triangle. 2. Prove that quadrilateral PLUS with the vertices P(2,1), L(6,3), U(5,5), and S(1,3) is a parallelogram. 29
3. 4. The vertices of quadrilateral ABCD are A(-3,-1), B(6,2), C(5,5) and D(-4,2). Prove that quadrilateral ABCD is a rectangle. 30
5. The vertices of quadrilateral ABCD are A(-3,1), B(1,4), C(4,0) and D(0,-3). Prove that quadrilateral ABCD is a square. 6. Quadrilateral METS has vertices M(-5, -2), E(-5,3), T(4,6) and S(7,2). Prove by coordinate geometry that quadrilateral METS is an isosceles trapezoid. 31